Abstract

This paper deals with point target detection in infrared images of the sky for which there are local variations of the gray level mean value. We show that considering a simple image model with the gray level mean value varying as a linear or a quadratic function of the pixel coordinates can improve mixed segmentation–detection performance in comparison to homogeneous model-based approaches.

Spectral density (averaged over the 64 square windows of 16×16=256 pixels shown in Fig. 1) of the difference between the pixel gray level and the mean value estimated in each window using the MC model (column 1), the ML model (column 2), and the MQ model (column 3). Row 1 is related to the image in Fig. 1a, row 2 to the image in Fig. 1b, and row 3 to the one in Fig. 1c.

Histograms of the difference between the pixel gray level and the mean value estimated in each square windows of 16×16=256 pixels shown in Fig. 1 using the MC model (bottom curve), the ML model (middle curve), and the MQ model (top curve), for the image in (a) Fig. 1a, (b) Fig. 1b, and (c) Fig. 1c. For comparison, the Gaussian PDF with the same mean and variance as the samples have been superimposed on the histograms (black dotted curve). For better visualization, the curves associated with the ML and the MQ models have been vertically translated from 0.01 and 0.02, respectively.

Evolution of the detection probability as a function of the Pfa for (a) a constant background with μΩr=2.1104 and σΩr=90 and, for (b) and (c), a linear background with αΩr=19, βΩr=−4.4, γΩr=2.0104, and σΩr=18, whose parameters correspond to typical parameter values estimated on the 16×16 pixel square windows in Fig. 1a. The region Ωr is a 11×11 pixel window and the target (with a 11dB SNR and a positive contrast) is not centered on the window but is located either d pixels upward (black curves) or d pixels downward (gray curves) from the window center. In (a) and (b), d=1 pixel and in (c), d=2 pixels. The target detection has been performed using either the MC model (continuous curves), the ML model (dashed curves) or the MQ model (dotted curves). The crossed curves in (b) and (c) are the average between the two curves obtained with the MC model. The plots have been obtained by averaging 105 samples.

Detection probability as a function of the region size for different Pfa and for square regions centered on the target location. The Pfa is fixed to (a) 10−3, (b) 10−4, and (c) 10−5, and the SNR of the target (which is centered on the window and has a positive contrast) is equal to 11dB (bottom curves) and 14dB (top curves). The plots have been obtained by averaging 105 samples and with background parameters set to μΩr=2.1104 and σΩr=90 when detecting with the MC model; αΩr=19, βΩr=−4.4, γΩr=2.0104, and σΩr=18 with the ML model; and aΩr=2.210−1, bΩr=7.210−2, cΩr=2.010−2, dΩr=−6.2, eΩr=−6.7, fΩr=2.1104, and σΩr=17 with the MQ model, which correspond to typical parameter values estimated on the 16×16 pixel square windows in Fig. 1a.

Detection probability as a function of the region size Nr for different Pfa (10−3, 10−4, 10−5) and for regions with (a) a vertical shape (with a 5 pixel width and a height h so that Nr=5h), (b) a square shape, and (c) a horizontal shape (with a 5 pixel height and a width w so that Nr=5w). The SNR of the target (which is centered on the window and has a positive contrast) is equal to 13dB and the plots have been obtained averaging 105 samples. The background parameters have been set to μΩr=2.1104 and σΩr=90 when detecting with the MC model; αΩr=19, βΩr=−4.4, γΩr=2.0104, and σΩr=18 with the ML model; and aΩr=2.210−1, bΩr=7.210−2, cΩr=2.010−2, dΩr=−6.2, eΩr=−6.7, fΩr=2.1104, and σΩr=17 with the MQ model, which correspond to typical parameter values estimated on the 16×16 pixel square windows in Fig. 1a.

Number of segmented regions containing fewer than N pixels as a function of N, averaged on the three images in Fig. 1 using the MC (solid black line), ML (continuous gray line), or MQ (dashed black line) model.

Evolution of the detection probability as a function of the Pfa for four mixed segmentation–detection techniques applied to 64 infrared images extracted from the same sequence as the image in Fig. 1a. The SNR of the targets is set to 11dB (with a positive contrast). The false alarm and detection probabilities have been estimated in the 431×46 pixel portion of the sky shown in Fig. 1a (dashed gray rectangle).

Insertion of a 13dB target (with a positive contrast) in the sky of the infrared image in Fig. 1b. (a) Zoom around the target location. Zoom of the segmentation results obtained on this image using (b) the MC model, (c) the ML model, and (d) the MQ model.

GLRT values ρi0j0(U) obtained in Fig. 10a. A target is detected if ρi0j0(U)>ηΩr,i0j0(U), and ηΩr,i0j0(U) is shown by the dashed line so that the Pfa=10−4. These values have been plotted for the line that contains the target, using (a) the MC model, (b) the ML model, and (c) the MQ model for both segmentation and detection and (d) using the MC model for segmentation and the ML model for detection. The target is located at column number 131.